Bitcoin clients can technically store information related to the blockchain in any format it so chooses. From a technical perspective, a bitcoin client will store information related to the blockchain in some kind of database. The important part of the blockchain is the unspent outputs, and there is a limit to how large this can get due to the limited number of bitcoin and the smallest allowable fraction of a bitcoin (satoshi). Even if the blockchain were to become too large to fit on a single hard drive, it could potentially be stored on multiple hard drives, and the majority of the data would never need to be accessed after it is validated one time.

There is also a technical limit as to how large the blockchain can be as a function of the block number, as each block has a maximum size.